Algorithms for Black-Box Fields and their Application to Cryptography

extended abstract
  • Dan Boneh
  • Richard J. Lipton
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1109)


We introduce the notion of a black box field and present several algorithms for manipulating such fields. Black box fields arise naturally in cryptography and our algorithms have several cryptographic implications. First, our results show that any algebraically homomorphic cryptosystem can be broken in sub-exponential time. The existence of such cryptosystems was posed as an open problem in [12]. Second we show that over elliptic (or hyperelliptic) curves the hardness of computing discrete-log implies the security of the Diffie-Hellman protocol. This provable security of the Diffie-Hellman protocol over elliptic curves demonstrates an additional advantage of elliptic curve cryptosystems over conventional ones. Finally, we prove that manipulating black box fields over the rationals is as hard as factoring integers.


Elliptic Curve Finite Field Elliptic Curf Hyperelliptic Curve Elliptic Curve Cryptosystems 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Dan Boneh
    • 1
  • Richard J. Lipton
    • 1
  1. 1.Princeton UniversityPrinceton

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